Until the fall of 2009, YouTube.com (hereafter, YouTube), the third most trafficked website in the world1, used a user-generated rating system. In the final essay, we apply the concepts of quantitative response equilibrium (QRE) and cursed equilibrium (CE) behavior to data generated in the laboratory.
Introduction
Literature Review
Unlike their model, players in our game receive one of many different signals about the quality of a product; players can send one of many messages, unlike the binary messages of voting; and purchase decisions for players acting in the second period must satisfy incentive compatibility, whereas the choice decision in a voting model is made according to a fixed rule. The tension in the model comes from the fact that players who send messages also have costs, a facet that has also been explored in the voting literature (Palfrey and Rosenthal 1983) (Ledyard 1984).
Model, Notation, and the Game
Primitives
Nature chooses the state of the world before the first period begins, and we assume that each state is equally likely. Furthermore, we assume that a sender's posterior belief about the probability of the high state of the world is increasing in a player's draw value.
Strategies and Equilibrium
In equilibrium, conditional on the state of the world, any two players have the same beliefs about the actions of other (third-party) players (regardless of types). Since P r(θ|vi) is common knowledge and P r(θ|tr) is known in equilibrium, we can extend Lemma 2.3.7 to say that every player of the same type also has the same beliefs.
Theory
The Receiver
From the proof of Theorem 2.4.1 we can see that if it is not the case that E[vr|θl]<. Thus, when considering informational equilibria, we can assume distributions with such expectations without loss of generality.
Senders
Where the form of the probability of conditional events follows from Lemma 2.3.7 and where the posterior beliefs of player i follow from Assumption 2.3.4. Indeed, optimal communication strategies must balance the player's certainty about the state of the world with his marginal expected utility in each state of the world.
Equilibrium
Another way to interpret Lemma 2.4.6 is to fix some v0 around which the recipient partition of the type space is formed. In this interpretation, Lemma 2.4.6 constrains the participation probabilities, requiring that the ratio of participation probabilities conditional on the state of the world must be such that v0=vΦ.
One Sender
Messaging
This simple equilibrium messaging rule leads to Proposition 2.5.2, which tells us that although the set of messaging strategies supported in equilibrium can be large, the form of the messaging strategy is irrelevant as long as there are at least 3 unique messages. For any equilibrium of any game in which N = 1, there exists an outcome-equivalent equilibrium of the same game in which each type of sender sends one of only three (or fewer) messages.
Participation
For any equilibrium of any game in which N = 1, there exists an outcome-equivalent equilibrium of the same game in which each type of sender sends one of only three (or fewer) messages. B) If g(θh|µΦ)> ν, then. The intuition is simple for all cases of Lemma 2.5.3: a sender will send a message if and only if the net benefit of sending his best available message outweighs the cost of doing so.
Equilibrium
Thus, as in the Crawford and Sobel model, in one-sender game equilibria, the sender's value space is partitioned into convex subsets within which the sender sends outcome-equivalent messages. In terms of our conjectures, this implies that in the one-sender case we reject both honest and exaggerated conjectures (the latter due to Proposition 2.5.2) and accept the extreme participation conjecture.
Multiple Senders
- An Extended Example
- The Mean
- The Median
- Thumbs Up or Thumbs Down
For our example, the relative posterior belief about the state of the world is shown in Figure 2.1.b as a function of v1. As is clear, the value of the sender becomes more certain that the state of the world is the high state.
Discussion and Conclusion
We showed that participation in all equilibria is increasing in the extremism of a sender's value on at least part of the value domain, and we identify conditions under which the extreme participation conjecture holds for the entire domain. An aggregation mechanism cannot do better than letting the receiver see each individual assessment, because the receiver can best infer the state of the world by observing the sender's signals directly.
Introduction
Related Experiments
The "winner's curse" describes the phenomenon in which the winner of a total value auction discovers that the value of the item he won is less than he expected (perhaps because he did not fully understand the information contained in the fact, that if he won, he had the highest private value of the item). The "curse of the swing voter" similarly describes the phenomenon in which a rational voter, given that his vote swings in an election, prefers to abstain, even if he prefers one candidate to another. Bazerman and Samuelson showed in a series of experiments that the winner's curse prevails in a laboratory setting, Ball et al. 1991) showed in a similar but repeated experiment that only a small proportion of subjects learn to avoid the winner's curse.
The Model
These articles confirm the comparative statics of their theoretical inspiration – that greater divergence in the interests of senders and receivers results in less information transmission, but also shows significant overtransmission of information on the part of the senders and significant overconfidence in the interests of the channels. messages from recipients.
Theory
Messaging
The analog of the left-hand side of inequality IC.1 and the right-hand side of inequality IC.2 to the ratio of marginal effects of the optimal message in each state of the world should be clear (one can add both the numerator and the denominator toM (t0i)−M(ti)). Considering the case in which vi0 > vi >v, the intuition here is that for any type, a sender increases its message until the ratio of the marginal effect of its message matches as closely as possible the ratio of its posterior, given the discreteness of the message space; any larger (smaller) and the effect of his message in the low (high) state will exceed the effect of his message in the high (low) state to an extent that outweighs his certainty that the state is high (low) .
Equilibrium
The proof of Proposition 3.4.3 shows that incentive compatibility inequalities imply that the components driving the comparative statics E[u|(vi, ci), ρi= 1] share X(vi). Proposition 3.4.3 tells us that in the environment and equilibria considered here, we accept the extreme participation assumption: participation increases as types move away from the neutral signal ˆv. The incentive compatibility conditions of Lemma 3.4.1 constrain the communication strategies that can be expected in equilibrium, but not enough to speak of an exaggeration assumption.
One Sender
The final case can form part of an equilibrium if and only if the strategies of that equilibrium also form an equilibrium when aΦ= 0.5. Theorem 3.4.4 says that there can only be an equilibrium in which the receiver mixes upon receiving the null message if the strategies that make up that equilibrium are also an equilibrium when the receiver mixes equally between the two products. Theorem 3.4.4 says that, without significant loss of generality, we can limit our search for equilibria to the three cases in which aΦ.
Experimental Design
N of the subjects in each group were randomly selected and assigned the role of sender. This draw was described to participants as a "clue to the quality of the new product". At the end of the first treatment, the experimenter described the change in the number of transmitters for the second part of the experiment.
Experimental Results
One Sender
This can be seen in the black solid line in Figure 3.5b, which represents average participation rates. Additionally, Figure 3.6 shows the frequency at which each message was sent as a function of the sender's signal. This appearance is made more dramatic by Figure 3.5b, which shows senders' average participation rates as a function of their signal.
Two and Five Senders
The best answer to empirical play in the five-sender treatment is increasingly puzzling. This reflects that the receiver makes the correct choice more often (relative to expected value) in the processing of five senders. Error rates were highest (in terms of first-order stochastic dominance) in the five-sender treatments and lowest in the one-sender treatment.
Discussion and Conclusion
In fact, in the five-sender treatment, for two types, the best response was to attenuate (toward the central message, away from sincerity) their signal. Deviations in the relative probabilities of events are the main factor in forming the best response correspondences. Cost density used in laboratory experiments .. a) Probability of participation, conditional on the signal of a sender.
Introduction
CE-QRE approximates well the comparative statics of observed messages across treatments, however we find that the model substantially under-predicts the extent of sincere messages occurring in the data. In the next section, we discuss quantity response equilibria (QRE), CE, and CE-QRE and how the concepts relate to the game presented in Chapter 2. In the appendices we first derive the expected utility in a fully cursed equilibrium and then present code used to generate estimates for our models.
Theory
CE-QRE
In this case, we calculated the fully cursed expected utility of each action as well as the Bayesian expected utility.
Experimental Results
One of the most striking aspects of the figures and of Table 4.3 is the variance in the maximum likelihood estimate of ˆχ across roles and treatments. So every level of curse less than fully cursed will result in the same best response. We estimate ˆλ to be lower in the one-transmitter treatment than in the five-transmitter treatments, but because of the different estimates of χ,.
Discussion and Conclusion
Therefore, by the definition of the pdf of the order statistic (and using the change of variable). Assume that the acceptor and players 2 through N play strategies according to the statement of the proposition. Finally, we want to show that the assumed strategy for the receiver is the best response to the sender's strategies.
This is the value of the aggregated message that excludes the sender's message, resulting in the aggregated message being exactly M(ˆv)1. The game is exactly the same as in the first part of the experiment.
Chapter 2
The probability of a given joint message was then the sum of the joint message probabilities given the level of participation, weighted by the probability of that level of participation. The idea of the functions was to calculate the probability of each permutation of messages sent in each country of the world in a computationally modest way. It worked by going through the set of feasible messages for each participant, keeping track of the sum of all messages sent and the product of the probabilities of those messages being sent in each condition.
Chapter 3
Chapter 4
